Locking-free numerical methods for nearly incompressible elasticity and incompressible flow on moving domains

Considerable effort is invested in the development of meshless methods under the claim that such methods have superior performance as compared to standard finite elements. This claim is often justified by poor performance of finite elements in situations where the mesh undergoes large distortion, and by a better ability of meshless methods to deal with the incompressibility constraint. In this paper these claims are investigated on a series of problems in incompressible elasticity and incompressible fluid flow. A standard displacement formulation and a mixed formulation with stabilization to circumvent the LBB-condition are used. The equations are integrated using stabilized nodal integration as well as Gauss quadrature. In the displacement formulation nodal integration effectively removes locking, while in the mixed formulation no stabilization is required. Nodal integration on median-dual cells is preferred over Voronoi cells due to computational efficiency. On very irregular nodal sets finite element and Sibson shape functions perform equally well, both when using Gauss quadrature as well as nodal integration.